Logik für Informatiker Logic for computer scientists

Transcription

1 Logik für Informatiker for computer scientists WiSe 2011/12

2 Language, proof and logic LPL book detailed introduction into first-order logic with many exercises Boole construct truth tables Tarski s world evaluate logical formulas within a blocks world Fitch construct proofs Grinder gives automatic feedback to your solutions requires purchase of the CD (ca. 13 EUR) or the book (ca. 25 EUR, with CD) Platform for exercises: logic.informatik.uni-bremen.de also reachable via

3 The formal language PL1 PL1 is the formal language of first-order predicate logic Why do we need a formal language? Slides from Prof. Barbara König, Universität Duisburg-Essen folien.pdf

4 The language of PL1: individual constants Individual constants are symbols that denote a person, thing, object Examples: Numbers: 0, 1, 2, 3,... Names: Max, Claire Formal constants: a, b, c, d, e, f, n1, n2 Each individual constant must denote an existing object No individual constant can denote more than one object An object can have 0, 1, 2, 3... names

5 The language of PL1: predicate symbols Predicate symbols denote a property of objects, or a relation between objects Each predicate symbol has an arity that tell us how many objects are related Examples: Arity 0: Gate0 is low, A, B,... Arity 1: Cube, Tet, Dodec, Small, Medium, Large Arity 2: Smaller, Larger, LeftOf, BackOf, SameSize, Adjoins... Arity 3: Between

6 The interpretation of predicate symbols In Tarski s world, predicate symbols have a fixed interpretation, that not always completely coindices with the natural language interpretation In other PL1 languages, the interpretation of predicate symbols may vary. For example, may be an ordering of numbers, strings, trees etc. Usually, the binary symbol = has a fixed interpretation: equality

7 Atomic sentences in propositional logic (Boole): propositional symbols: a, b, c,... in PL1 (Tarski s world): application of predicate symbols to constants: Larger(a,b) the order of arguments matters: Larger(a,b) vs. Larger(b,a) Atomic sentences denote truth values (true, false)

8 Function symbols Function symbols lead to more complex terms that denote objects. Examples: father, mother +, -, *, / This leads to new terms denoting objects: father(max) 3*(4+2) mother(father(max)) This also leads to new atomic sentences: Larger(father(max),max) 2<3*(4+2)

9 al validity; satisfiability A sentence A is a logically valid, if it is true in all circumstances. A sentence A is a satisfiable, if it is true in at least one circumstance. A circumstance is in propositional logic: a valuation of the atomic formulas in the set { true, false } in Tarski s world: a block world

10 Consequences...

11 al consequence A sentence B is a logical consequence of A 1,..., A n, if all circumstances that make A 1,..., A n true also make B true. In symbols: A 1,..., A n = B. A 1,..., A n are called premises, B is called conclusion. In this case, it is a valid argument to infer B from A 1,... A n. If also A 1,... A n are true, then the valid argument is sound.

12 al consequence A sentence B is a logical consequence of A 1,..., A n, if all circumstances that make A 1,..., A n true also make B true. In symbols: A 1,..., A n = B. A 1,..., A n are called premises, B is called conclusion. In this case, it is a valid argument to infer B from A 1,... A n. If also A 1,... A n are true, then the valid argument is sound.

13 al consequence examples All men are mortal. Socrates is a man. So, Socrates is mortal. (valid, sound) All rich actors are good actors. Brad Pitt is a rich actor. So he must be a good actor. (valid, but not sound) All rich actors are good actors. Brad Pitt is a good actor. So he must be a rich actor. (not valid)

14 al consequence examples All men are mortal. Socrates is a man. So, Socrates is mortal. (valid, sound) All rich actors are good actors. Brad Pitt is a rich actor. So he must be a good actor. (valid, but not sound) All rich actors are good actors. Brad Pitt is a good actor. So he must be a rich actor. (not valid)

15 al consequence examples All men are mortal. Socrates is a man. So, Socrates is mortal. (valid, sound) All rich actors are good actors. Brad Pitt is a rich actor. So he must be a good actor. (valid, but not sound) All rich actors are good actors. Brad Pitt is a good actor. So he must be a rich actor. (not valid)

16 al consequence examples All men are mortal. Socrates is a man. So, Socrates is mortal. (valid, sound) All rich actors are good actors. Brad Pitt is a rich actor. So he must be a good actor. (valid, but not sound) All rich actors are good actors. Brad Pitt is a good actor. So he must be a rich actor. (not valid)

17 al consequence examples All men are mortal. Socrates is a man. So, Socrates is mortal. (valid, sound) All rich actors are good actors. Brad Pitt is a rich actor. So he must be a good actor. (valid, but not sound) All rich actors are good actors. Brad Pitt is a good actor. So he must be a rich actor. (not valid)

18 al consequence examples All men are mortal. Socrates is a man. So, Socrates is mortal. (valid, sound) All rich actors are good actors. Brad Pitt is a rich actor. So he must be a good actor. (valid, but not sound) All rich actors are good actors. Brad Pitt is a good actor. So he must be a rich actor. (not valid)